Well-Posedness and Porosity for Symmetric Optimization Problems

In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a σ-porous set.

Influence of Silica Fumes on Compressive Strength and Wear Properties of Glass Ionomer Cement in Dentistry

BACKGROUND Glass ionomer cements (GIC) are an interesting restorative option due to their biocompatibility. However, it has limitations that challenge its survival in oral environment due its porous set matrix influencing the properties of the cement. This study was conducted to evaluate the influence of the addition of varying concentrations of silica fumes (SF) on the properties of GIC by field emission scanning electron microscopy [FESEM] and energy-dispersive spectroscopy [EDX]. The final set matrix of GIC remains porous, compromising the mechanical properties, limiting its extended use clinically. Incorporation of silica fumes, a pozzolan, as an additive in GIC serves as a potential filler by increasing its compressive strength and reducing wear properties. METHODS The cement was divided into 5 groups based on the absence or presence of varying concentrations (0.5, 1, 1.5, 2 %) of silica fumes; conventional glass ionomer group (CG) (I) and 0.5, 1, 1.5, 2 silica fumes incorporated glass ionomer cement (SG) (II, III, IV & V) respectively. Compressive strength and wear resistance were subjected to Universal Testing Machine and Pin on Tribometer respectively. The microstructure and the elemental composition of prepared specimens of all the groups were evaluated using FESEM and EDX. Data obtained was analysed using Statistical Package for the Social Sciences (SPSS) V22.0 (IBM, USA) followed by one-way analysis of variance (ANOVA) and post hoc Tukey test (P < 0.05). RESULTS Except 0.5SG, increased compressive strength and decreased wear of glass ionomer material was observed as the concentration of silica fumes increased. Of all the concentrations, 2SG had significantly increased compressive strength (221.62 ± 22.84 MPa) compared to CG (167.38 ± 36.94 MPa) (P < 0.05). Significantly increased resistance to wear was noted in 2SG (11.80 ± 2.58 µm) compared to CG (20.40 ± 2.07 µm) (P < 0.05). The set matrix of silica fumes modified GIC showed minimal / absence of pores with dispersion of crystalline particles as the concentration of SF increased. EDX revealed similar constitution of minerals but, varied with increased concentration of silica fumes. CONCLUSIONS 2 % silica fumes incorporated glass ionomer cement (2SG) enhanced the properties of conventional glass ionomer cement. KEY WORDS Compressive Strength, EDX, Field Emission Scanning Electron Microscope, Glass Ionomer Cement, Silica Fumes, Pozzolan

Porous subsets in the space of functions having the Baire property

The comparison of some subfamilies of the family of functions on the real line having the Baire property in porosity terms is given. We prove that the family of all quasi-continuous functions is strongly porous set in the family of all cliquish functions and that the family of all cliquish functions is strongly porous set in the family of all functions having the Baire property.We prove also that the family of all Świątkowski functions is lower 2/3-porous set in the family of cliquish functions and the family of functions having the internally Świątkowski property is lower 2/3-porous set in the family of cliquish functions.

Comparison of Some Subfamilies of Functions Having The Baire Property

We prove that the family Q of quasi-continuous functions is a strongly porous set in the space Ba of functions having the Baire property. Moreover, the family DQ of all Darboux quasi-continuous functions is a strongly porous set in the space DBa of Darboux functions having the Baire property. It implies that each family of all functions having the A-Darboux property is strongly porous in DBa if A has the (∗)-property.

Self-affine sets with fibred tangents

We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a rotation${\mathcal{O}}$such that all tangent sets at that point are either of the form${\mathcal{O}}((\mathbb{R}\times C)\cap B(0,1))$, where$C$is a closed porous set, or of the form${\mathcal{O}}((\ell \times \{0\})\cap B(0,1))$, where$\ell$is an interval.

Some properties of porouscontinuous functions

The notion of porouscontinuous function will be introduced on the base of porous set and relations between porouscontinuous, continuous and quasicontinuous functions will be investigated.

Fr échet Differentiability of Real-Valued Functions

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the corresponding point of Fréchet differentiability may be found outside any given σ‎-porous set. This new result considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ‎-porous sets is new even in the Lipschitz case. The chapter first discusses the use of variational principles to prove Fréchet differentiability before analyzing a one-dimensional mean value problem in relation to Lipschitz functions. It shows that results on existence of points of Fréchet differentiability may be generalized to derivatives other than the Fréchet derivative.

Porosity and ε‎-Fr échet differentiability

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.

Unavoidable Porous Sets and Nondifferentiable Maps

This chapter discusses Γ‎ₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a σ‎-porous set whose complement is null on all n-dimensional surfaces and the multidimensional mean value estimates fail even for ε‎-Fréchet derivatives. Previous chapters have established conditions on a Banach space X under which porous sets in X are Γ‎ₙ-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into n-dimensional spaces hold. This chapter investigates in what sense the assumptions of these main results are close to being optimal.